Due to the short settling times of metals in DA white dwarf atmospheres, any
white dwarfs with photospheric metals must be actively accreting. It is
therefore natural to expect that the metals may not be deposited uniformly on
the surface of the star. We present calculations showing how the temperature
variations associated with white dwarf pulsations lead to an observable
diagnostic of the surface metal distribution, and we show what constraints
current data sets are able to provide. We also investigate the effect that
time-variable accretion has on the metal abundances of different species, and
we show how this can lead to constraints on the gravitational settling times.Comment: 4 pages, 5 figures, accepted for publication in the Astrophysical
  Journal Letters, updated reference

Montgomery, M. H.

Thompson, S. E.

von Hippel, T.

English

arXiv

Due to the short settling times of metals in DA white dwarf atmospheres, any white dwarfs with photospheric metals must be actively accreting. It is therefore natural to expect that the metals may not be deposited uniformly on the surface of the star. We present calculations showing how the temperature variations associated with white dwarf pulsations lead to an observable diagnostic of the surface metal distribution, and we show what constraints current data sets are able to provide. We also investigate the effect that time-variable accretion has on the metal abundances of different species, and we show how this can lead to constraints on the gravitational settling times

Montgomery, Michael H.

Thompson, S.E.

von Hippel, Ted

Embry-Riddle Aeronautical University

Constraining the Surface Inhomogeneity and Settling Times of Metals on Accreting White Dwarfs

Due to the short settling times of metals in DA white dwarf atmospheres, any white dwarfs with photospheric metals must be actively accreting. It is therefore natural to expect that the metals may not be deposited uniformly on the surface of the star. We present calculations showing how the temperature variations associated with white dwarf pulsations lead to an observable diagnostic of the surface metal distribution, and we show what constraints current data sets are able to provide. We also investigate the effect that time-variable accretion has on the metal abundances of different species, and we show how this can lead to constraints on the gravitational settling times.National Science Foundation AST-0507639, AST-0607480Delaware Asteroseismic Research CenterAstronom

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L133The Astrophysical Journal, 685: L133–L136, 2008 October 1  2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.CONSTRAINING THE SURFACE INHOMOGENEITY AND SETTLING TIMES OF METALSON ACCRETING WHITE DWARFSM. H. Montgomery,1,2 S. E. Thompson,2,3 and T. von Hippel1,4Received 2008 July 24; accepted 2008 August 20; published 2008 September 9ABSTRACTDue to the short settling times of metals in DA white dwarf atmospheres, any white dwarfs with photosphericmetals must be actively accreting. It is therefore natural to expect that the metals may not be deposited uniformlyon the surface of the star. We present calculations showing how the temperature variations associated with whitedwarf pulsations lead to an observable diagnostic of the surface metal distribution, and we show what constraintscurrent data sets are able to provide. We also investigate the effect that time-variable accretion has on the metalabundances of different species, and we show how this can lead to constraints on the gravitational settling times.Subject headings: accretion, accretion disks — convection — diffusion — stars: oscillations —stars: variables: other — white dwarfsOnline material: color figures1. ASTROPHYSICAL CONTEXTThere are two main classes of white dwarf stars: those withhydrogen-rich atmospheres (spectral type DA) and those withhelium-rich atmospheres (non-DA spectral types). The reasonfor this is that the high surface gravities of white dwarfs leadto efficient gravitational settling, with the lightest elements ris-ing to the surface. In addition, some DA white dwarfs havespectra showing metal lines of elements such as Ca and Mg(Zuckerman et al. 2003), and these stars are referred to asDAZs; about 20% of all DAs fall into this category. Recently,Dufour et al. (2007) have announced a new class of whitedwarfs with carbon-dominated atmospheres, the “hot DQ”stars, several examples of which have been found in the SloanDigital Sky Survey (Liebert et al. 2003).The presence of metals in the DAZs is intriguing since thesettling timescale for the metals may be many orders of mag-nitude shorter than the evolutionary age of these objects. In-deed, for DAZs with K, the settling timescale canT ∼ 12,000effbe on the order of days or weeks, meaning that these objectsare experiencing ongoing accretion (Koester & Wilken 2006).This ongoing accretion is consistent with the fact that nearlya dozen of these objects have detected dust disks (“debrisdisks”; see Tokunaga et al. 1990; Becklin et al. 2005; Kilic etal. 2005; Reach et al. 2005; Kilic et al. 2006; von Hippel etal. 2007; Jura et al. 2007a, 2007b; Farihi et al. 2008), and thesedisks are assumed to be the sources of the metal lines seen inthese white dwarf atmospheres. The best-studied object of thisDAZ class with an observed disk, G29-38, is also a multiper-iodic variable white dwarf (DAV), pulsating in nonradialg-modes with periods of a few hundred to 1000 s.The technique of asteroseismology uses the observed pul-sation modes of a star to infer and constrain the interior struc-ture of the star, thus obtaining information on the star’s structureas a function of radius (Bradley & Winget 1994; Kawaler &Bradley 1994; Metcalfe et al. 2002; Montgomery et al. 2003).1 Department of Astronomy, University of Texas at Austin, Austin, TX;mikemon@astro.as.utexas.edu.2 Delaware Asteroseismic Research Center, Mt. Cuba Observatory, Green-ville, DE.3 Department of Physics and Astronomy, University of Delaware, Newark,DE.4 Department of Physics, Siena College, Loudonville, NY.In contrast to this, our approach in this Letter is to use thedifferent angular dependence of the pulsations to constrain theaccretion process. Since the accretion is most likely occurringthrough a disk, it is natural to suppose that the metals may notbe uniformly distributed across the star’s surface. We presentcalculations showing how we can place constraints on the non-uniformity of the accretion process.2. GRAVITATIONAL SETTLING AND HORIZONTAL DIFFUSIONFor all our calculations, we use the DAZ G29-38 as ourtemplate, since, given its brightness, it has the greatest potentialfor successful measurement of a surface inhomogeneity. Inaddition, we have archival data on this star appropriate to thisapplication. In this section, we will therefore attempt to quantifythe importance of gravitational settling and horizontal diffusionassuming a model with parameters similar to those of G29-38.Bergeron et al. (2004) find K andT p 11,820 log g pefffor this star, while Koester et al. (1997) find8.14 T peffK and . Interpolating in the tables of Koes-11,600 log g p 8.05ter & Wilken for Ca yields a settling time of ∼13 days for thefirst set of parameters5 and a settling time of ∼23 days for thesecond. We note that while unseen helium in the atmospherecould lengthen these settling times considerably (e.g., Garcı́a-Berro et al. 2007), its presence is inconsistent with the pulsationresults for this star: such an amount would imply a much deepersurface convection zone, in conflict with that found by Mont-gomery (2005).Since these stars should have surface convection zones, thedominant form of horizontal transport of Ca will be due to theturbulent viscosity. We can estimate the size of this diffusioncoefficient as , where is a typical convective velocityD ∼ v l vhC Cand is the assumed “mixing length” for convection. Fromlhour white dwarf evolution code (e.g., see Montgomery et al.1999), we find for both sets of stellar parameters that D ≈cm2 s1.101.5 # 10For these simple exploratory calculations we assume azi-muthal symmetry for both the accretion and the surface metaldistribution, i.e., and , whereZ p Z(v, t) S p S(v, t) Z pis the metal abundance, is the source functionZ(v, t) S p S(v, t)5 Due to a misreading of Table 2 in Koester & Wilken (2006), von Hippel& Thompson (2007) calculated a settling time of ∼7 days.L134 MONTGOMERY, THOMPSON, & VON HIPPEL Vol. 685Fig. 1.—Possible surface metal distributions for accretion centered on thepoles (dashed curve) or the equator (solid curve) as a function of the polarangle (colatitude). [See the electronic edition of the Journal for a color versionof this figure.]of metals accreting onto the white dwarf, v is the “colatitude”of a point on the star’s surface, and t is time.Since convection will uniformly mix material vertically, wecan treat the convective region as a single “zone” and write anequation for the time evolution of Z as a function of v and t:Z(v, t) 2p gZ(v, t)  D∇ Z(v, t)  S(v, t), (1)htwhere is the settling rate, is the hor-2g { 1/(settling time) ∇hizontal part of the Laplacian operator, and the other variablesare as defined above. An estimate of the relative importanceof sinking to spreading is , where is the radius2h { gR /D R∗ ∗of the white dwarf: using the Bergeron et al. (2004)h ∼ 50values whereas for those of Koester et al. (1997). Whenh ∼ 30, the metals will sink before they have a chance to diffuseh k 1horizontally, while for , the metals will have a chanceh K 1to mix thoroughly horizontally before sinking, producing anearly uniform surface distribution.In Figure 1 we show the metal distributions which arise fromsolutions of equation (1). The dashed curve is the equilibriumdistribution which results from constant accretion centered atthe poles, and the solid curve is that which results from constantaccretion centered on the equator.3. THE DIAGNOSTICThe flux variations observed in pulsating white dwarfs aredue almost entirely to temperature changes on the surface ofthe stars (Robinson et al. 1982). These same temperaturechanges will also affect the equivalent widths (EWs) of anyspectral lines and, in particular, the EWs of metal lines. Thesemetals may not be uniformly distributed across the star’s sur-face, and since the temperature variations are also nonuniform,we hope to be able to constrain the surface metal distribution.The relevant diagnostic we have developed, denoted by R,is the ratio of the fractional EW amplitude to the fractionalflux amplitude:dAEWS/AEWS AEWR { p , (2)dF /F AX X Fluxwhere is the fractional amplitude of EW variations andAEWis the amplitude of the fractional flux variations observedAFluxin the given passband X.To compute R, we assume a particular surface temperatureperturbation of the form . We then use modeldT/T ∝ Y (v, f)matmospheres to turn this into EW and flux variations on thesurface of the star, taking into account the fact that the EW ofthe Ca lines will be directly proportional to the local abundanceof Ca (e.g., the curves in Fig. 1). For the passband X we assumea wavelength response appropriate to the Argos CCD with aBG40 filter on the 2.1 m telescope at McDonald Observatory(Nather & Mukadam 2004); the wavelength range is taken tobe 3000–7000 , with a peak response at 5400 . Finally, we˚ ˚A Aintegrate the result across the visible surface of the star. Limbdarkening is automatically taken into account by this procedure.We note for a uniform distribution of metals that R pfor and for . Thus, even by mea-2.71  p 1 R p 2.59  p 2suring R for a single mode it may be possible to tell whethermetals are uniformly distributed on the star’s surface. To obtainfurther constraints, we need R determinations from other modesand/or additional information such as the inclination angle ofthe star.In Figure 2 we show the R diagnostic as a function of in-clination angle . The different curves are labeled accordingvito their and m values as . The left-hand plot is for the F mSpolar distribution of metals given in Figure 1, and the right-hand plot is for the equatorial distribution. The thin horizontalboxed region shows the value of R expected if the metal dis-tribution is uniform; we see that a nonuniform distribution isvery unlikely to produce a value of R in this range.4. COMPARISON WITH OBSERVATIONSIn 1996 Clemens & van Kerkwijk obtained over 4 hr oftime-resolved spectroscopy of the DAV G29-38 (Clemens etal. 2000; van Kerkwijk et al. 2000; Clemens et al. 1999). Whilenot originally intended for this purpose, we can use this as anexample data set for the technique proposed in the previoussection. We take the amplitudes for the EW variations fromthe analysis of von Hippel & Thompson (2007), and we usethe amplitudes of the broadband (5200–5500 ) flux variationsÅas determined by van Kerkwijk et al. (2000).6 We calculatefor each of the modes, taking into account theR p A /AEW Fluxerrors on all quantities.We show the results of this procedure in Figure 3. Of thesix modes we identified, only the first two have R values withsmall enough error bars to provide any constraint on the Cadistribution. The 614 s mode provides the most convincingevidence for a nonuniform Ca distribution, since its R value isa full 3 j above the range produced by a uniform distribution.The 818 s mode may also provide some evidence, although itis only about 1.3 j above the value expected from a uniformdistribution.With only one statistically significant value of R, we cannothope to infer anything further about the Ca distribution. For-tunately, we do have additional information. Clemens et al.(2000) used time-resolved spectroscopy to determine that thismode has . In addition, Montgomery (2005), by mod- p 1eling the nonlinear pulse shape of this mode, was able to further6 Even though this wavelength range is much narrower than that used inthe previous section for Fig. 2, the central wavelengths of the two passbandsare nearly the same and the results obtained are virtually indistinguishablefrom one another.No. 2, 2008 METAL INHOMOGENEITY AND SETTLING IN PULSATING WDS L135Fig. 2.—The diagnostic R as a function of inclination angle . The left panel shows the results for the polar distribution of metals shown in Fig. 1, and theviright panel shows the results for the equatorial distribution. The different curves are labeled by the values of the relevant pulsation modes. [See the electronicF mSedition of the Journal for a color version of this figure.]Fig. 3.—Values of R for modes in the DAV G29-38 as obtained from 4 hrof Keck data in 1999. The values for the first six highest amplitude modesare shown, with the period in seconds indicated above each point. Only thefirst two modes have a signal-to-noise ratio large enough to be useful.Fig. 4.—Comparison of the observed R value for the 614 s mode in G29-38 (shaded region) to the value expected for a polar, a uniform, and an equa-torial distribution of Ca, as a function of inclination angle . The data areviconsistent only with the equatorial case.specify that it is an , mode. With just this one p 1 m p 1additional constraint, we are somewhat surprisingly able toconstrain our models of the Ca distribution. Figure 4 showsthe R values derived using an equatorial, a polar, and a uniformdistribution, for a variety of inclination angles. The 614 s mea-surement is inconsistent with a polar distribution of metals andis most consistent with an equatorial distribution.5. TIME-VARIABLE ACCRETION AND SETTLINGIn the case of G29-38, accretion is thought to be from adebris disk having roughly asteroidal composition, so that sev-eral elements are being accreted simultaneously. In particular,if we look at Ca and Mg, these two elements have differentsettling rates, with Mg settling about 50% more slowly thanCa.Accretion is not expected to be a completely steady process.Von Hippel and Thompson (2007) found the EW of Ca lines inG29-38 to vary on a timescale of weeks to years, with additionalevidence for variations on shorter timescales (T. von Hippel 2008,private communication). On the other hand, Debes & López-Morales (2008) found the observed EW variations of G29-38to be consistent within the measurement errors. For the presentwe treat this as an open question and ask what effect time-variable accretion would have on the measured atmosphericabundances of different chemical species and how this can beused as a probe of the settling process in these stars.For this problem it is sufficient to consider the sphericallyaveraged version of equation (1):dZ (t)i p g Z (t)  X S(t), (3)i i idtwhere is the abundance of element i, is its settling rate,Z gi iand (assumed to be constant) is the relative fraction of el-Xiement i in the accreted material.In Figure 5 we illustrate the solution of equation (3) for aparticular set of parameters and a given accretion rate. Theaccretion rate assumed is shown in the top panel of Figure 5.L136 MONTGOMERY, THOMPSON, & VON HIPPEL Vol. 685Fig. 5.—Bottom panel: The fractional relative abundance of element 1( day1; solid line) and element 2 ( day1; dashed line)g p 1/13 g p 1/261 2as given by the solution of eq. (3). Top panel: The time-variable accretionrate (arbitrary units) assumed for these calculations. [See the electronic editionof the Journal for a color version of this figure.]For the two elements we have assumed that day1g p 1/131and day1. In the bottom panel we plot the abun-g p 1/261dance of each element divided by its mean value, i.e., its “frac-tional” abundance: the solid curve is that of element 1, and thedashed curve is that of element 2. By using relative/fractionalabundances, we have divided out the effect of the parameter.XiThe first feature to note is that the abundances of the twoelements are highly correlated; i.e., when one goes up or down,so does the other. In addition, we see that the amplitudes ofthe excursions of element 1 are larger than those of element2. In fact, taking the standard deviations of each of the curves,we find that that of element 1 is about 1.8 times that of element2. In other words, the element with the shorter settling timeexperiences larger fractional variations, and these variationsmay be used for a rough estimate of the ratio of the settlingtimes. Thus, periodic observations of a white dwarf with morethan one metal line may be able to place constraints on therelative settling times of different chemical species in whitedwarf atmospheres.6. CONCLUSIONSWe have shown how the temperature variations due to stellarpulsation can be used to constrain the metal distribution on thesurface of a white dwarf, and we have shown that data currentlyin hand for the star G29-38 suggest that the metal distribution,and therefore the accretion, may be equatorial. If further studiessupport an equatorial distribution for Ca in G29-38, this arguesagainst magnetic accretion onto spots near the poles. Non-magnetic equatorial accretion would further imply that the inneredge of the disk is much thinner than the white dwarf’s radius.A physically thin disk is consistent with observations to date(e.g., Jura et al. 2007a; von Hippel et al. 2007).We have also shown how the observed variations in atmo-spheric abundance of two chemical species can be used to placeconstraints on their settling rates. The element with the fastersettling rate will show the larger fluctuations, and, at leastwithin the context of this simplified model, the ratio of themagnitude of these excursions from the mean will be approx-imately proportional to the settling rate.Thus, the time variability of these systems, both in terms ofthe pulsations of the white dwarf and in terms of the variableaccretion rate, allows us to probe the physics of accretion andgravitational settling in these objects.M. H. M. and T. v. 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Constraining the Surface Inhomogeneity and Settling Times of Metals on Accreting White Dwarfs

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